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Pre-Semi-Closed Sets and Pre-Semi- Separation Axioms in Intuitionistic Fuzzy
Topological Spaces
A. Bhattacharjee1 and R.N. Bhaumik2
1Department of Mathematics, D.D.M College, Khowai, Tripura West-799201 E-mail: [email protected]
2Rtd. Professor of Mathematics, Tripura University, Suryamaninagar, Tripura-799022
32 Dhaleswar, Assam-Agartala Road, Near Joyguru Pharmacy, Agartala-College, Tripura, India, 799004
E-mail: [email protected]
Abstract
The aim of this paper is to introduce and study different properties of pre-semi closed sets in intuitionistic fuzzy topological spaces. As applications to pre-semi- closed sets we introduce pre-semi T1/2-spaces, semi- pre T1/3 space and pre-semi T3/4-spaces and obtain some of their basic properties.
Keywords: Intuitionistic Fuzzy (IF) sets, IF semi closed set, IF semi-pre (=β) closed set, IF generalized closed set, IF regular closed set, IF pre-semi closed set, IF pre- semi T1/2 space, IF semi- pre T1/3 space, IF pre- semi T3/4 space, etc.
1 Introduction
The concept of intuitionistic fuzzy set was introduced by Atanasov [1] in 1983 as a generalization of fuzzy sets. This approach provided a wide field to the
generalization of various concepts of fuzzy mathematics. In 1997 Coker [3]
defined intuitionistic fuzzy topological spaces. Recently many concepts of fuzzy topological space have been extended in intuitionistic fuzzy(IF) topological spaces. Murugesan and Thangavelu [6] intrduced the concept of pre-semi- closed sets in fuzzy topological spaces. In the present paper we introduce and study different properties of pre-semi-closed sets, pre-semi T1/2-spaces, semi- pre T1/3 space and pre-semi T3/4-spaces in IF topological spaces.
2 Preliminaries
Definition 2.1 Let X denotes a universe of discourse. Then a fuzzy set A in X is defined as a set of ordered pairs A= {< x, µA(x) > : x ∈ X },where µA(x): X → [0, 1] is the grade of belongingness of x into A. Thus the grade of non belongingness of x into A is equal to 1−µA(x). However ,while expressing the degree of membership of any given element in a fuzzy set, the degree of non membership is not always expressed as a complement to1.Therefore Atanassov [1,2] suggested a generalization of fuzzy set, called an intuitionistic fuzzy set. In the present paper intuitionistic fuzzy will be denoted by IF only.
An IF set in X is given by a set of ordered triples A = {< x, µA(x),νA(x)>: x
∈X},where µA(x),νA(x) : X → [0,1] are functions such that 0≤µA(x) +νA(x)≤ 1,∀ x
∈ X. The numbers µA(x) and νA(x) represent the degree of membership and degree of non-membership for each element x∈X to A ⊂ X, respectively.
Definition 2.2[2] Let A and B beIF sets of the form A= {< x, µA(x),νA(x)> : x ∈ X }and B= { < y,µB(y) ,ν B(y) > : y ∈ Y}. Then
(a) A ⊆ B if and only If µA(x) ≤µB(x) and νA(x) ≥νB(x) for all x∈ X.
(b) Ac = {< x,νA(x), µA(x) > : x ∈ X }.
Definition 2.3
[10]
Two IF sets A and B are said to be quasi-coincident, denoted by A q B if and only if there exists an element x ∈ X such that µA(x) > νB(x) or νA(x)<µB(x).The expression ‘not quasi-coincident’ will be abbreviated as A q B.
Proposition 2.1
[10]
For any two IF sets A and B of X, A q B if and only if A ⊆ Bc.Definition 2.4[5] An IF set A of an IF topological space (X,τ ) is said to be IF semi closed set if int( cl (A)) ⊆ A.
Definition 2.5[5] An IF set A of an IF topological space (X,τ ) is said to be IF semi-pre (=β)-closed set if int( cl( int (A))) ⊆ A.
Definition 2.6[9] An IF set A of an IF topological space (X,τ ) is said to be IF generalized closed if cl(A) ⊆ O whenever A ⊆ O and O is IF open (X,τ ).
An IF set A of an IF topological space (X,τ) is said to be IF generalized open if its complement Ac is IF generalized closed.
Proposition2.2[9] Every IF open set is IF generalized open but its converse may not be true.
Notation 2.1 Let (X,τ ) be an IF topological space. Then the family of IF regular (respectively semi-pre) closed sets in X, may be denoted by r (respectively sp).
Definition2.7[7] Let A be an IF set in an IF topological space (X,τ ). Then IF semi pre interior and semi pre closure of A is denoted by spint (A) and spcl(A), defined by
spint (A) = ∪ {G:G⊆ A ,G is IF semi pre open set in X } spcl(A) = ∩{B:A⊆B,B is IF semi pre closed set in X }.
Definition 2.8[7] An IF set A of an IF topological space ( X,τ ) is said to be IF generalized semi-pre closed set if spcl(A) ⊆ O whenever A ⊆ O and O is IF open set in (X,τ ).
An IF set A of an IF topological space (X,τ) is said to be IF generalized semi-pre open if its complement Ac is IF generalized semi-pre closed.
Definition 2.9[7] An IF topological space (X,τ ) is said to be IF semi-pre T1/2
space if every IFGSP closed set in X is an IF semi-pre closed set in X.
3 Pre-Semi-Closed Sets
Definition 3.1 Let A be an IF set in an IF topological space (X,τ ). Then A is called an IF pre-semi closed set in X if spcl(A) ⊆O whenever A ⊆ O and O is IF generalized open set in X.
Example 3.1 Consider the IF topological space (X,τ ),where X={a, b} and τ ={0∼,
1∼,U}, U = <x,(a/.9,b/.2),(a/.1,b/.8)>. Since every IF open set is IF g-open so 0∼, 1∼, U are IF generalized open sets. Let A = <x,(a/.7,b/.2),(a/.3,b/.8)> be an IF set in X. Then A is an IF pre-semi-closed set in X, for if A ⊆ O and O is IF generalized open set in X, then
O = 1∼ and hence spcl(A) ⊆ O.
Theorem 3.1 Every IF semi-pre-closed set in an IF topological space (X,τ ) is IF pre-semi closed set.
Proof. Let A be an IF semi-pre closed set in an IF topological space ( X,τ ).
Suppose that A ⊆ O and O is IF generalized open set in X. Since A is an IF semi- pre closed set, hence spcl(A) = A. Thus spcl(A) = A ⊆ O, and hence A is IF pre- semi closed set.
But the converse may not true as shown in the following example.
Example 3.2: Consider the IF topological space (X,τ ), where X={a, b} and τ = {0∼, 1∼,U}, U = <x,(a/.7,b/.3),(a/.3,b/.7)>. Since every IF open set is IF generalized open so 0∼, 1∼, U are IF g-open sets. Let A = <x,(a/.8,b/.3),(a/.2,b/.7)>
be an IF set in X. Then A is an IF pre-semi-closed set in X, for if A ⊆O and O is IF generalized open set in X, then
O =1∼ and hence spcl(A) ⊆ O. But A is not an IF semi-pre-closed set, for int(A) = U, so cl(int(A)) = 1∼⊃ A.
Remark 3.1 The IF pre-semi closed ness is independent from IF generalized closed ness as shown in the following two examples.
Example 3.3 In example 3.1 A is an IF pre-semi-closed set in X. But not an IF generalized closed set, for A ⊆ U and U is IF open set in X, but cl(A) = 1∼⊄ U.
Example 3.4 Consider the IF topological space (X,τ ), where X={a, b} and τ
={0∼, 1∼,U}, U = <x,(a/1,b/0),(a/0,b/.5)>. Let A = <x,(a/1,b/.2),(a/0,b/.3)> be an IF set in X. Then A is an IF generalized closed set in X but not an IF pre-semi- closed set in X.; for if A ⊆ A and A is an IF generalized open set in X, but int(cl(int(A))) = 1∼ ⊃ A and hence spcl(A) = 1∼ ⊃ A.
Theorem 3.2 Every IF pre-semi closed set in an IF topological space (X,τ ) is IF generalized semi pre closed.
Proof. Let A be an IF pre-semi closed set in an IF topological space ( X,τ ).
Suppose that A ⊆ O and O is an IF open set in X, then spcl(A) ⊆ O and O is IF generalized open set in X. Hence A is an IF generalized semi pre closed in X.
But the converse may not true as shown in the following example.
Example 3.5 In example 3.4 A is an IF generalized semi pre closed set in X but not an IF pre-semi closed set.
Remark 3.2 Every IF semi pre closed set is IF generalized semi pre closed set but its converse may not be true [7]. Hence the relationships of IF semi pre closed set, IF pre-semi closed set, IF generalized semi pre closed sets are as follows.
IF semi pre closed set ⇒ IF pre-semi closed set ⇒ IF generalized semi pre closed set.
However the converses are not true in general.
Theorem 3.3 Let A be an IF set in an IF topological space ( X,τ ). Then A is an IF pre-semi closed if and only if (A q F) ⇒ (spcl(A) qF) for every IF generalized closed set F of X.
Proof.
Necessity Let F is an IF generalized closed set of X and (A q F). Then by proposition 2.1 A ⊆ Fc and Fc is IF generalized open in X. Now since A is IF pre- semi closed, spcl (A) ⊆ Fc. Hence (spcl(A) qF).
Sufficiency Let O is an IF generalized open set of X such that A⊆ O that is A ⊆ (Oc)c. Hence by proposition 2.1 (A q Oc) and Oc is IF generalized closed set in X.
Hence by hypothesis (spcl(A) q Oc). Therefore spcl(A) ⊆ (Oc)c. That is spcl(A)
⊆ O. Hence A is IF pre-semi closed in X.
Lemma 3.1 Let A be an IF set in an IF topological space (X,τ ). Then A∪ int(cl(int(A))) ⊆ spcl(A).
Theorem 3.4 Let A be an IF set in an IF topological space (X,τ ). If A is IF generalized open and IF pre-semi closed, then A is IF semi-pre closed.
Proof. Since A is IF generalized open and IF pre-semi closed, it follows that A ∪ int(cl(int(A))) ⊆ spcl(A) ⊆ A. Hence int(cl(int(A))) ⊆ A and A is IF semi-pre closed.
Theorem 3.5 Let A be an IF set in an IF topological space (X,τ ). Then the following are equivalent:
(i) A is IF regular open.
(ii) A is IF open and IF pre-semi closed.
(iii) A is IF open and IF generalized semi pre closed set.
Proof (i) ⇒ (ii) Let A be an IF regular open set in an IF topological space ( X,τ ).
Then A is both IF open and IF semi closed. Now since every IF semi closed set is IF semi pre closed set, hence by theorem 3.1 A is an IF pre-semi closed set.
(ii) ⇒ (iii) Let A be IF open and IF pre-semi closed. Then by theorem 3.2 A is IF generalized semi pre closed set.
(iii) ⇒ (i) Let A be IF open and IF generalized semi pre closed set. Then A ⊆ A and A is an IF open set in X and so A ∪ int(cl(int(A))) ⊆ spcl(A) ⊆ A. Hence int(cl(int(A))) ⊆ A. Since A is IF open it follows that int(cl(A)) ⊆ A = int(A) ⊆ int(cl(A)). Hence A is IF regular open.
Lemma 3.2 Let A be an IF set in an IF topological space (X,τ ). Then spcl(
spcl(A)) = spcl(A).
Theorem 3.6 Let A be an IF pre-semi closed set in an IF topological space (X,τ ).
If B is an IF set in X such that A ⊆ B ⊆ spcl(A),then B is also IF pre-semi closed.
Proof. Let B be an IF set in an IF topological space (X,τ ) such that B⊆ O and O is an IF generalized open set in X. Then A ⊆ O, since A is IF pre-semi closed, hence by lemma 3.2 spcl(B) ⊆ spcl( spcl(A)) = spcl(A) ⊆ O. Hence, B is an IF pre-semi closed in X.
Definition 3.2 An IF set A in an IF topological space (X,τ ) is called an IF pre- semi open if and only if its complement Ac is IF pre-semi closed.
Theorem 3.7 An IF set A in an IF topological space (X,τ ) is called an IF pre- semi open if F ⊆ spint(A) whenever F ⊆ A and F is IF generalized closed set in X.
Proof. Let an IF set A in an IF topological space (X,τ ) is IF pre-semi open and F is an IF generalized closed set in X such that F ⊆ A. Then Ac ⊆ Fc, where Ac is IF pre-semi closed and Fc is an IF generalized open set in X. Hence from definition 3.1 spcl(Ac) ⊆ Fc. Hence (Fc )c ⊆ (spcl(Ac))c. That is F ⊆ spint(Ac)c = spint(A).
Theorem 3.6 Let A be an IF pre-semi closed set in an IF topological space (X,τ ).
If B is an IF set in X such that spint(A) ⊆ B ⊆ A, then B is also IF pre-semi open.
Proof Let B be an IF set in an IF topological space (X,τ ) such that F ⊆ B and F is IF generalized closed set in X. Then F ⊆ A, since A is IF pre-semi closed, hence F ⊆ spint(A). Therefore F ⊆ spint(A) ⊆ B. Hence, B is an IF pre-semi open in X.
4 Pre-Semi-Separation Axioms
Definition 4.1 An IF topological space (X,τ ) is said to be IF pre- semi T1/2 space if every IF pre-semi closed set in X is IF semi-pre closed set in X.
Theorem 4.1 Every IF semi-pre T1/2 space is an IF pre- semi T1/2 space.
Proof. Let (X,τ ) be an IF semi-pre T1/2 space and A be an IF pre-semi closed set in X. Now by theorem 3.2 every IF pre-semi closed set is an generalized semi pre closed set in X. Hence, A is an generalized semi pre closed set. in X and consequently A is IF semi-pre closed set in X. Thus (X,τ ) is IF pre- semi T1/2
space.
Definition 4.2 An IF topological space (X,τ ) is said to be IF semi- pre T1/3 space if every generalized semi pre closed set in X is IF pre- semi closed set in X.
Theorem 4.2 Every IF semi-pre T1/2 space is an IF semi- pre T1/3 space.
Proof. Let (X,τ ) be an IF semi-pre T1/2 space and A be an generalized semi pre closed set in X. Then, A is IF semi-pre closed set in X. Now by theorem 3.1 every IF semi-pre-closed set in X is IF pre-semi closed. Thus (X,τ ) is IF semi- pre T1/3
space.
Definition 4.3 An IF topological space (X,τ ) is said to be IF pre- semi T3/4 space if every IF pre-semi closed set in X is IF pre closed set in X.
Theorem 4.3 Every IF pre- semi T3/4 space is an IF pre- semi T1/2 space.
Proof. Let ( X,τ ) be an IF pre- semi T3/4 space and A be an IF pre-semi closed set in X. Then A is IF pre closed set in X. Now every IF pre closed set in X is IF semi-pre closed set in X. Hence, A is an IF semi-pre closed set in X. Thus ( X,τ ) is IF pre- semi T1/2 space.
References
[1] K. Atanasov, Intuituionistic fuzzy sets, ITKR,s session, Sofia, VII(1983) (in Bulgaria).
[2] K. Atanasov, Intuituionistic fuzzy sets, Fuzzy Sets and Systems, 20(1986), 87-96.
[3] D. Coker, An introduction to IFT spaces, Fuzzy Sets and Systems, 88(1) (1997) 81-89.
[4] M. Caldas and R.K. Saraf, Preserving Fuzzy sg-Closed Sets, Proyeccions, 20(2) (2001), 127-138.
[5] H. Gurcay, D. Coker and A.H. Es, On fuzzy continuity in intuitionistic fuzzy topological spaces, The Journal of Fuzzy Mathematics,5(2)(1997), 365-378.
[6] S. Murugesan, P. Thangavelu, Fuzzy pre-semi-closed sets, Bulletin of the Malayasian Mathematical Sciences Society, (2) 31(2) (2008), 223-232.
[7] R. Santhi, and D. Jayanthi, Intuituioistic fuzzy generalized semi-pre closed sets(accepted)
[8] S.S. Thakur and J. Bajpai, Intuituionistic fuzzy sg-continuous mappings, International Journal of Applied Mathematical Analysis and Applications, 5(1) (2010),45-51.
[9] S.S. Thakur and R. Chaturvedi, Generalized closed set in intuitionistic fuzzy Topology, The Journal of Fuzzy Mathematics, 16(3) (2008)559-572.
[10] S.S. Thakur and R. Chaturvedi, Regular generalized closed sets in intuitionistic fuzzy topologicalspaces, Studii Si Cercetari Stiintifice, Universitateta Din Bacau, Seria: Matematica, 16(2006), 252-272.
